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Part III
Part III
In this, the final part of the course, we will introduce the notions of local and global viewpoints of
number theory, which began with the notion of p-adic numbers. (p as usual denote a rational prime.)
The basic idea is that many problems in number theory can be treated by looking at solutions mod
m. We saw, say with the example of x2 + y 2 , that we can rule out any number ≡ 3 mod 4 being of
the form x2 + y 2 .
On the other hand, suppose, for a given n, we knew a mod m solution to
x2 + y 2 ≡ n mod m
for each m. Hasse’s idea was that if these local solutions (solutions mod m for each m) are “sufficiently
compatible,” then we can paste them together to actually construct a global solution in Z. In fact
it suffices to consider the cases where m = pe is a prime power. What it means for the solutions to
be sufficiently compatible means is the following. Consider x2 + y 2 = 244. A solution to this in Z
would mean in particular we have solutions mod 2 and mod 4. Here are two:
x2 + y 2 ≡ 12 + 12 ≡ 0 mod 2, x2 + y 2 ≡ 02 + 22 ≡ 0 mod 4
In the mod 2 solution x, y must both be odd, but in the mod 4 solution both x and y are even, so
there is no way to paste together these local solutions to get a solution in integers, hence we say
they are not compatible.
Essentially what the p-adic integers Zp are, are the elements of
(Z/pZ) × (Z/p2 Z) × (Z/p3 Z) × · · ·
which are compatible in the above sense. In other words, a p-adic integer x = (xn ) gives a congruence
class xn mod pn for each n such that xn+1 ≡ xn mod pn . We can form the field of fractions of the
p-adic integers to obtain the field of p-adic number Qp . The advantage of this is we can use field
theory, which is much stronger than ring theory, whereas we couldn’t do this with a single Z/pn Z,
since Z/pn Z doesn’t embed in a field as there are nontrivial zero divisors (unless n = 1). (Even
though Zp “contains” all of these Z/pn Z’s, it turns out to be an integral domain.)
After discussing the p-adic numbers, we will discuss applications to quadratic forms in several
variables. This naturally leads into the topic of modular forms, which is slated to be taught next
year, and we will not discuss them in any detail here.
We will follow this with an introduction to adéles, which is considered a global way of studying
things. Just like the p-adic numbers put together information mod pn for all n, the adèles AQ put
together Qp for all p. Moreover, one can do all this over an arbitrary number field. Namely, for
any number field K and prime ideal p of OK , one can define the field Kp of p-adic numbers. Then
one define the adéles AK of K, which is essentially a product of all the Kp’s. It turns out that AK
provides an alternative way to study the class group ClK as well as class field theory, which studies
the abelian extensions of K.
The adelic picture is important for several reasons, not least of which is it allows for a vast
generalization of class field theory, known as Langlands’ program, or non-abelian class field theory.
As a special case, Langlands’ program (together with Wiles’ famous work) includes the famous
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Taniyama–Shimura(–Weil) correspondence between elliptic curves and modular forms, which is famous for proving Fermat’s last theorem. While (abelian) class field theory is more or less considered
a closed book now (which is of course not to say that everything is known about abelian extensions), the Langlands’ program is only in a toddler stage, and lies at the heart of the research of
several faculty members here. The Langlands’ program and the generalized (or grand) Riemann
hypothesis are the two most important outstanding problems in both number theory and the theory
of automorphic forms/representations.
Time permitting, we will give a brief introduction to abelian class field theory and Langlands’
program. The second semester of next year’s Modular Forms course should contain a more detailed
introduction to Langlands’ program.
6
p-adic numbers
Throughout this chapter p will denote a fixed prime number of N.
In the introduction to Part III, we briefly described the p-adic integers are elements in
(an ) ∈ (Z/pZ) × (Z/p2 Z) × (Z/p3 Z) × · · ·
which are compatible in the sense that the natural map Z/pn+1 Z → Z/pZ maps an+1 to an . There
are several different ways to describe the p-adic numbers, which were first introduced by Hensel
at the end of the 1800’s. Before we proceed into the formalities of the p-adic numbers, it may be
interesting to describe Hensel’s original viewpoint of the p-adic numbers.
The basic idea came from an analogy with algebraic geometry. The basic premise of modern
mathematics is that to study some object X, it is helpful to study functions on X. In particular,
to study the complex numbers C, one may choose to study the polynomial ring C[x]. (The space
C is the set of points, and the ring C[x] is called the coordinate ring of C.) One of the early
observations in complex algebraic geometry was that the set of maximal ideals of C[x] is just the
set of (principal) ideals generated by a linear polynomial of the form x − p0 for some point p0 ∈ C.
In other words, there is a bijection between C and the maximal ideals of C[x], given by a point
p0 ∈ C corresponds to the ideal of all polynomials which vanish at p0 . Further if f (x) ∈ C[x], then
we have the map
C[x] → C[x]/(x − p0 ) � C
f (x) �→ f (p0 ),
i.e., to mod out by a maximal idea (x − p0 ) in C[x], just means substituting in x = p0 for a
polynomial f (x) ∈ C[x], i.e., this “mod (x − p0 )” map C[x] → C just sends a polynomial f (x) to its
value at a point p0 .
Now instead, let’s try to imagine Z in place of C[x] as a coordinate ring. What should the
space of points be? Well, in analogy with the above, a good candidate is the set of maximal ideals
of Z, i.e., the set of all nonzero prime ideals (p) of Z. In other words, if we consider the space
P = {pZ : p ∈ N} of points as the natural number primes, then the coordinate ring “P[x],” i.e.,
the “polynomials on the space P,” are just the integers n ∈ Z. How do we evaluate a “polynomial”
n ∈ Z on a point p ∈ P? Just consider the map
Z → Z/pZ
n �→ n mod p.
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In other words, the analogue of polynomials in one variable over C, when we replace C with the set
of primes P, are the functions on P given by integers n ∈ Z such that n(p) = n mod p. One obvious
difference is that for any p0 ∈ C, the space C[x]/(x − p0 ) � C, so all functions in the coordinate
ring C[x] really map into C. But in the case of p ∈ P, the spaces Z/pZ are all non-isomorphic, so
it’s harder to think of n(p) = n mod p ∈ Z/pZ as a function, since its image lands in a different
space (Z/pZ) for each p.
To go further with this analogy, one can ask about a notion of derivatives of the functions
n(p) = n mod p. Observe for a polynomial f (x) ∈ C[x], we can always write f (x) in the form
f (x) = a0 + a1 (x − p0 ) + a2 (x − p0 )2 + · · · + ak (x − p0 )k
where each ai ∈ C, and the m-th derivative at p0 is just given by m!am . Similarly, for any n ∈ Z,
we can write n in the form
n = a0 + a1 p + a2 p2 + · · · + ak pk
where 0 ≤ ai < p, so the m-th derivative at p should be m!am .
Since power series are such a powerful tool in function theory, Hensel wanted to apply the techniques of power series to number theory. If we work with more general functions than polynomials
in C[x], namely analytic functions at p0 , we can write them as power series about x = p0
f (x) = a0 + a1 (x − p0 ) + a2 (x − p0 )2 + · · · ∈ C[[x]] (ai ∈ C).
Analogously, we can consider formal power series in a prime p ∈ P given by
n = a0 + a1 p + a2 p2 + · · · ∈ Zp (0 ≤ ai < p).
These formal power series are the p-adic integers Zp . (Note Zp contains Z by just restrict to finite
sums, i.e., “polynomials” in p.)
Even more generally than analytic functions at p0 , one often considers meromorphic functions
on C which may have a pole (go to infinity) at p0 , e.g., the Riemann zeta function ζ(s) has a pole at
s = 1. These functions still have a series expansion at p0 , but it needs to start with some negative
power of x − p0 . These are called Laurent series, and explicitly are of the form
f (x) = a−k (x − p0 )−k + a1−k (x − p0 )1−k + · · · + a0 + a1 (x − p0 ) + a2 (x − p0 )2 + · · · ∈ C((x)) (ai ∈ C).
Analogous to this, one can take formal power series in p with coefficients between 0 and p with a
finite number of negative terms
n = a−k p−k + a1−k p1−k + · · · + a0 + a1 p + a2 p2 + · · · ∈ Qp (0 ≤ ai < p),
and this will give us the p-adic numbers Qp . (Note Qp contains all rational numbers with denominator a power of p.)
This analogy may seem a little far fetched, and you might wonder if Hensel had one too many
beers at this point, but the usefulness of the p-adic numbers allows us to recognize his ideas as
brilliant, as opposed to crazy talk. We summarize the analogy in the table below, though to fully
appreciate it, one should be familiar with complex function theory. Nevertheless, even if you are
not, it may be helpful to refer back to this table after learning more about Zp and Qp .
We can now explain why Zp and Qp are called local objects, specifically, local rings and local
fields. A power or Laurent series expansion of some function f (x) around a point p0 may only
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Table 3: Complex functions vs.
C—space of points
C[x]—polynomials over C
f (x) = a0 + a1 (x − p0 ) + · · · ak (x − p0 )k
functions
� analytic at p0i
f (x) = ∞
i=0 ai (x − p0 )
functions�
meromorphic at p0
i
f (x) = ∞
i=−k ai (x − p0 )
p-adic numbers
P = {p}—set of primes
Z—“polynomials” over P
n = a0 + a1 p + · · · + ak pk
Zp —p-adic
� integers
i
n= ∞
i=0 ai p
Qp —p-adic
� numbers
i
n= ∞
i=−k ai p
converge nearby p0 , even though the function may be defined (as a meromorphic function) on all of
C. Since a power series is essentially meaningless outside its radius of convergence, power series in
general only give local information about functions f (x) (namely, near p0 ). Similarly, the elements
of Zp and Qp will give “local” information about the prime p ∈ P.
The main references I will be using for this chapter are [Neukirch] and [Serre], as these were
the books I originally learned the theory from, though most if not all of this material may be
found in many books on algebraic number theory, and of course any book specifically on p-adic
numbers, of which there are a few. There is also a nice analytic/topological presentation in
[Ramakrishnan–Valenza], which leads into adèles.
6.1
Definitions
Fix a prime p ∈ N.
Definition 6.1.1. The set of p-adic integers, denoted Zp , are the formal power series of the form
∞
�
i=0
ai pi = a0 + a1 p + a2 p2 + · · · , 0 ≤ ai < p.
�∞
Observe the series i=0 ai pi converges if and only if it is finite, i.e., if ai = 0 for all i > k for
some k. In this case, this finite sum is an integer, and we can get any non-negative integer this way.
Accordingly we will view N ∪ {0} ⊆ Zp .
We can abbreviate this representation as an “infinite” base p representation of a “number:”
∞
�
i=0
ai pi = · · · a2 a1 a0
Note if the series is in fact finite, then this really is the base p representation of the corresponding
integer:
ak ak−1 · · · a2 a1 a0 = a0 + a1 p + a2 p2 + · · · + ak pk .
��
�
�
base p
Naively, we can think of a p-adic integer x as just a sequence (ai )∞
0 of numbers between 0 and p, but
the p-adic numbers will have more structure than just this. We define addition and multiplication
on Zp by just extending the usual addition and multiplication on base p representations of positive
integers.
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�
�∞
i
2
i
Example 6.1.2. Let x = 1 + 4 · 5 + 3 · 52 , y = ∞
i=0 1 · 5 ∈ Z5 , and z = 4 + 2 · 5 +
i=3 4 · 5 . We
can compute the sums
· · · 0000341
x
+ · · · 1111111
y
· · · 1112002 x + y
· · · 0000341
x
+ · · · 4444104
z
· · · 0000000 x + z
and we can compute a product
· · · 1111111
y
+ · · · 4444104
z
· · · 1110220 y + z
· · · 1111111 1 × · · · 1111111
+ · · · 4444440 40 × · · · 1111111
+ · · · 3333300 300 × · · · 1111111
· · · 4444401
x·y
Since x + z = · · · 00000, we may identify x with 396 and z with −396 in Z5 .
It is easy to see in general, that for any x ∈ Zp , the additive inverse of x (the additive zero is of
course · · · 00000 ∈ Zp ) also lies in Zp . Hence we may regard Z ⊆ Zp .
Exercise 6.1. Find the 7-adic representations for −7 and −121.
�
i
Exercise 6.2. Let x = 64 ∈ Z7 and y = 4 + 6 · 7 + ∞
i=2 2 · 7 ∈ Z7 . Compute x + y and x · y.
Exercise 6.3. What are the p-adic representations for −1 and
1
1−p
for arbitrary p?
Proposition 6.1.3. We have that Zp is a ring with Z as a subring.
The proof of this is elementary—it is just base p arithmetic with infinite sequences—but we will
see another justification for this from a more algebraic description below. The statement about Z
being a subring just means that with the identification of Z ⊆ Zp described above, the addition
and multiplication defined on Zp are compatible with those on Z, which is evident from the way we
defined them. Specifically, let φn denote the natural maps
φn
φ2
φ1
· · · −→ Z/pn+1 Z −→ Z/pn Z −→ · · · −→ Z/p2 Z −→ Z/pZ
Definition 6.1.4. The projective limit (or inverse limit) of Z/pn Z (with respect to φn ) as
n → ∞ is
�
�
�
lim Z/pn Z = (xn ) ∈
Z/pn Z : φn (xn+1 ) = xn for all n ≥ 1
←−
n
In other words an element (xn ) of lim Z/pn Z is a sequence of elements xn ∈ Z/pn Z which is
←−
compatible in the sense that xn+1 ≡ xn mod Z/pn Z. (Recall a direct or injective limit, written lim,
−→
is for when we have a sequence of objects which are successively included in each other. A projective
limit is for when we have a sequences of objects which are successive quotients (or projections) of
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each other. This is the natural way to construct an object X, in this case lim Z/pn Z, such that each
←−
Z/pn Z is a quotient of lim Z/pn Z. It’s of course not the smallest space X such that every Z/pn Z is
←−
a quotient of X—that would be Z—but it is certainly just as natural, if not more so. If you doubt
this, try to figure out how you could construct Z from the set of Z/pn Z’s.)
The reason for the compatibility requirements was already described in the introduction to Part
III. To state this reason a little differently,
� then idea was that we want to use Zp to study solutions
to equations in Z. If we just look at n Z/p Z, it’s not very meaningful. Note that any element
(xn ) ∈ Zp is the limit of integers xn ∈ Z, whereas a non-compatible
sequence is not a limit of
�
integers. For instance, the sequence (1, 2, 3, 0, 0, 0, . . .) in Z/pn Z
Proposition 6.1.5. We have a bijection
∞
�
n=0
Zp → lim Z/pn Z
←−
an pn �→ (sn )∞
n=1
where sn is the (image in Z/pn Z) of the n-th partial sum
sn = a0 + a1 p + a2 p2 + · · · + an−1 pn−1 .
From now on we use this bijection to identify Zp with lim Z/pn Z, and sometimes write our
←−
p-adic integers as formal power series expansions in p, and sometimes write them as sequences in
the projective limit of the Z/pn Z’s. There are some nice features of the projective limit approach.
First, there is a natural map from Z → Zp = lim Z/pn Z given by
←−
�
Z/pn Z
a �→ (a mod p, a mod p2 , a mod p3 , . . .) ∈
n
�
for any a ∈ Z. Further we can just define addition of and multiplication of elements of Z/pn Z.
Then it is immediate that Zp is a ring with Z as a subring, i.e., the proof of Proposition 6.1.3
is immediate. (We did not actually check that the two definitions of addition and multiplication
match, but this is certainly true when we restrict to the subring Z, since + and · are the standard
operations then. Since we can approximate any x ∈ Zp as a limit of xn ∈ Z, a density argument
shows + and · extend in a unique way to Zp , so the two definitions of + and · agree.)
Example 6.1.6. Suppose p = 2. Consider n = 75 ∈ Z. As a power series, we can write n =
1 · 1 + 1 · 2 + 1 · 23 + 1 · 26 . Alternatively, we can write
n = (1 mod 2, 3 mod 4, 3 mod 8, 11 mod 16, 11 mod 64, 75 mod 128, 75 mod 256, 75 mod 512, . . .)
as a sequence in the projective limit of Z/2n Z. Note in the projective limit version, it’s easier to
write down −n, namely
−n = (−1, −3, −3, −11, −11, −75, −75, −75, −75, . . .).
The usefulness of p-adic integers is that the precisely capture the answer of when an equation
is solvable mod pn for all n.
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Proposition 6.1.7. Consider a polynomial F (x1 , . . . , xk ) ∈ Z[x1 , . . . , xk ]. Then
F (x1 , . . . , xk ) ≡ 0 mod pn
is solvable for all n if and only if
F (x1 , . . . , xk ) = 0
is solvable in Zp .
Proof.
(⇐) Suppose we have a Zp -solution F (x,1 , . . . , xk ) = 0. Write xi = (xi1 , xi2 , xi3 , . . .) ∈
�
Z/pn Z. Then F (x1n , . . . xkn ) ≡ 0 mod pn for each n.
(⇒) Let x1n , . . . xkn ∈ Z/pn Z be a solution to F (x1n , . . . , xkn ) ∈ Z/pn Z for each n. One would
like to say that xi = (xin ) ∈ Zp , but the (xin )’s will not in general be compatible. Nevertheless,
we can construct a compatible sequence of solutions.
By the (infinite) pigeonhole principle, there is a (y11 , . . . , yk1 ) ∈ (Z/pZ)k such that
(x1n , . . . , xkn ) ≡ (y11 , . . . , yk1 ) mod p
for infinitely many n. Then F (y11 , y21 , . . . , yk1 ) ≡ 0 mod p since any of the (x1n , . . . , xkn ) above
give a solution mod p.
Similarly, there is a (y12 , . . . , yk2 ) ∈ (Z/p2 Z)k such that
(y12 , . . . , yk2 ) ≡ (y11 , . . . , yk1 ) mod p
and
(x1n , . . . , xkn ) ≡ (y12 , . . . , yk2 ) mod p2
for infinitely many n. Again we have F (y12 , y22 , . . . , yk2 ) �
≡ 0 mod p2 .
We continue this ad infinitum, and set yi = (yin ) ∈ n Z/pn Z, so in fact each yi ∈ Zp . Then
we have F (y1 , . . . , yk ) = 0 ∈ Zp since this expression must be 0 in each component of Zp =
lim Z/pn Z.
←−
In many cases, one can reduce checking the solvabilty of an equation mod pn to simply solvability
mod p. Here is a special case.
Lemma 6.1.8. (Hensel) Let f (x) ∈ Z[x], p a prime and n ∈ N. If p = 2, we assume n ≥ 2.
Suppose f (a) ≡ 0 mod pn for some a ∈ Z, but p � f � (a). Then for each n ≥ 1 there is an
b ∈ Z/pn+1 Z such that f (b) ≡ 0 mod pn+1 and b ≡ a mod pn .
Starting with n = 1 (or 2 if p = 2) applying this inductively, we see that if we have a root a of
a one-variable polynomial f (x) mod p (or mod 4), it lifts to a root an mod pn for�all n, provided
f � (a) �= 0. In fact, these roots an can be chosen to be compatible so that (an ) ∈ Z/pn Z lies in
Zp .
Here f � (x) is the formal derivative of f (x), in other words the derivative as a real polynomial.
Proof. The Taylor series for f (x) (regarded as a function of a real variable x) about x = a is
f (x) = f (a) + f � (a)(x − a) +
f �� (a)(x − a)2
f (d) (a)(x − a)d
+ ··· +
2!
d!
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where d is the degree of f (x). Suppose we take x of the form x = a + pn y. Then we have
f (x) = f (a) + f � (a)pn y +
f �� (a)p2n y 2
f (d) pdn y d
+ ··· +
2!
d!
By induction on j, it is easy to see for j ≥ 2 (or j ≥ 3 if p = 2) that pn+1 divides
words, we can have
f (x) ≡ f (a) + f � (a)pn y mod pn+1 .
pjn
j! .
In other
Since f (a) ≡ 0 mod pn , we can write f (a) = a0 pn so
f (x) ≡ a0 pn + f � (a)ypn ≡ (a0 + f � (a)y)pn mod pn+1 .
Since f � (a) is nonzero mod p, we can choose 0 ≤ y < p such that a0 + f � (a)y ≡ 0 mod p, so
f (x) ≡ 0 mod pn+1 and we can take b = x.
There are several ways in which one can generalize Hensel’s lemma, but we will not worry about
these here.
Exercise 6.4. Let a = a0 + a1 p + a2 p2 + · · · ∈ Zp . Show a is a unit in Zp if and only if a0 �= 0.
Exercise 6.5. Show Zp is an integral domain, i.e., there are no zero divisors.
Exercise 6.6. Show x2 = 2 has a solution in Z7 .
Exercise 6.7. Write
2
3
as a 5-adic integer.
Since Zp has no zero divisors, it has a field of fractions. By Exercise 6.4, we know the only
nonzero elements of Zp which are not invertible (w.r.t. multiplication) are the elements divisible by
p, hence the field of fractions is obtained by adjoining p1 to Zp , i.e., the field of fractions of Zp is
Zp [ p1 ]. Note that we can write the elements of Zp [ p1 ] uniquely in the form p−d a where a ∈ Zp and
d ≥ 0. If a = a0 + a1 p + a2 p2 + · · · , we can write
�
p−d a = a0 p−d + a1 p1−d + a2 p2−d + · · · =
a�n pn
(6.1)
n≥−d
where a�n = an+d . Thus we may define the p-adic numbers as formal series starting with some finite
negative power of p (called a formal Laurent series in p).
Definition 6.1.9. The p-adic numbers Qp is the set of formal Laurent series


�

Qp =
an pn : 0 ≤ an < p, d ≥ 0 .


n≥−d
We identify Qp with the field of fractions Zp [ p1 ] of Zp as in (6.1).
Exercise 6.8. Write
5
12
as a 2-adic number.
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6.2
Valuations
If R is an integral domain, a map | · | : R → R which satisfies
(i) |x| ≥ 0 with equality if and only if x = 0,
(ii) |xy| = |x||y|, and
(iii) |x + y| ≤ |x| + |y|
is called an absolute value on R. Two absolute values |·|1 and |·|2 are equivalent on R if |·|2 = |·|c1
for some c > 0. If we have an absolute value | · | on R, by (ii), we know |1 · 1| = |1| = 1. Similarly,
we know | − 1|2 = |1| = 1, and therefore | − x| = |x| for all x ∈ R.
Now a absolute value | · | on R makes R into a metric space with distance d(x, y) = |x − y|. (The
fact | − x| = |x| guarantees |y − x| = |x − y| so the metric is symmetric, and (iii) gives the triangle inequality.) Recall that any metric space is naturally embued with a topology. Namely, a basis of open
(resp. closed) neighborhoods around any point x ∈ R is given by the set of open (resp. closed) balls
Br (x) = {y ∈ R : d(x, y) = |x − y| < r} (resp. B r (x) = {y ∈ R : d(x, y) = |x − y| ≤ r}) centered at
x with radius r ∈ R.
Ostrowski’s Theorem says, that up to equivalence, every absolute value on Q is of one of the
following types:
| · |0 , the trivial absolute value, which is 1 on any non-zero element
| · |∞ , the usual absolute value on R
| · |p , the p-adic absolute value, defined below, for any prime p.
Here the p-adic absolute value defined on Q is given by
|x| = p−n
where x = pn ab with p � a, b. (Note any x ∈ Q can be uniquely written as x = pn ab where p � a, b
and ab is reduced.)
In particular, if x ∈ Z is relatively prime to p, we have |x| = 1. More generally, if x ∈ Z,
|x| = p−n where n is the number of times p divides x.
Note any integer x ∈ Z satisfies |x|p ≤ 1, and |x|p will be close to 0 if x is divisible by a high
power of p. So two integers x, y ∈ Z will be close with respect to the p-adic metric if pn |x − y for a
large n, i.e., if x ≡ y mod pn for large n.
Example 6.2.1. Suppose p = 2. Then
1
1
1
|1|2 = 1, |2|2 = , |3|2 = 1, |4|2 = , |5|2 = 1, |6|2 = , . . .
2
4
2
3
12
1 57
| |2 = 4, | |2 = , | |2 = 4.
4
17
4 36
With respect to | · |2 , the closed ball B 1/2 (0) of radius 12 about 0 is simply all rationals (in reduced
form) with even numerator. Similarly B 1/4 (0) of radius 14 about 0 is simply all all rationals (in
reduced form) whose numerator is congruent to 0 mod 4.
Exercise 6.9. Prove | · |p is an absolute value on Q.
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Recall, for a space R with an absolute value | · |, one can define Cauchy sequences (xn ) in R—
namely, for any � > 0, |xm − xn | < � for all m, n large. One forms the completion of R with respect
to | · | by taking equivalence classes of Cauchy sequences. Everyone knows that the completion of
Q with respect to | · |∞ is R. On the other hand, the completion of Q with respect to | · |p is Qp . To
see this, observe that
x0 = a−d p−d + a1−d p1−d + · · · + a0
x1 = a−d p−d + a1−d p1−d + · · · + a0 + a1 p
x2 = a−d p−d + a1−d p1−d + · · · + a0 + a1 p + a2 p2
..
.
1
gives a Cauchy sequence in Q with respect to | · |p . Precisely |xn+1 − xn |p = |an+1 pn+1 |p = pn+1
(unless xn+1 = xn , in which case it is of course 0). Hence these are Cauchy sequences, and their
limits are just formal Laurent series in Qp . Hence Qp is contained in the completion of Q with
respect to | · |p . It is also not hard to see that any Cauchy sequence in Qp converges (convince
yourself).
Hence, the Qp ’s are an arithmetic analogue of R, just being completions of the absolute values
on Q (Q is already complete with respect to the trivial absolute value—Q is totally disconnected
with respect to | · |0 ). This approach to constructing Qp gives both an absolute value and a topology
on Qp , which are the most important things to understand about Qp .
Precisely, write any x ∈ Qp as
x = am pm + am+1 pm+1 + · · · , am �= 0
for some m ∈ Z. Then we define the p-adic (exponential) valuation∗ (or ordinal) of x to be
ordp (x) = m.
Then
|x|p = p−m = p−ordp (x) .
Proposition 6.2.2. Zp = {x ∈ Qp : ordp (x) ≥ 0} = {x ∈ Qp : |x|p ≤ 1}. In particular Zp is a
closed (topologically) subring of Qp .
Proof. This is clear since
Zp =

�

an pn
n≥0
so Zp is precisely the set of x ∈ Qp with ordp (x) ≥ 0.



,
Corollary 6.2.3. The group of units Z×
p of Zp is
Z×
p = {x ∈ Qp : ordp (x) = 0} = {x ∈ Qp : |x|p = 1} .
∗
One often calls absolute values | · | valuations on a field. Thus sometimes there is a question of whether one
means the exponential valuation or the absolute value by the term “valuation.” For clarity, we will reserve the term
valuation for exponential valuation, and always refer to our absolute values as absolute values.
Even the term exponential valuation is somewhat confusing, as the exponential valuation is really the negative
logarithm − logp | · | of the absolute value. “The exponent valuation” might be clearer terminology.
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Proof. This is immediate from Exercise 6.4.
60
Exercise 6.10. Let p = 5. Determine ordp (x) and |x|p for x = 4, 5, 10, 217
150 , 79 . Describe the (open)
1
ball of radius 10 centered around 0 in Qp .
Exercise 6.11. Let x ∈ Q be nonzero. Show
|x|∞ ·
�
p
|x|p = 1.
This result will be important for us later.
Despite the fact that R and Qp are analogous in the sense that they are both completions of nontrivial
absolute values on Q, there are a couple of fundamental ways in which the p-adic absolute value
and induced topology are different from the usual absolute value and topology on R.
Definition 6.2.4. Let | · | be an absolute value on a field F . If |x + y| ≤ max {|x|, |y|}, we say | · |
is nonarchimedean. Otherwise | · | is archimedean.
The nonarchimedean triangle inequality, |x + y| ≤ max {|x|, |y|}, is called the strong triangle
inequality.
Proposition 6.2.5. | · |∞ is archimedean but | · |p is nonarchimedean for each p.
Proof. Everyone knows | · |∞ or Q or R is archimedean—this is what we are use to and the proof is
just |1 + 1|∞ = 2 > 1 = max {|1|∞ , |1|∞ }.
Now let’s show | · |p is nonarchimedean on Q. Since Q is dense in Qp (Qp is the completion of
Q), this will imply | · |p is nonarchimedean on Qp also. Let x, y ∈ Q. Write x = pm ab , y = pn dc ,
where a, b, c, d are relatively prime to p, and m, n ∈ Z. Without loss of generality, assume m ≤ n.
Then we can write
�a
c�
ad + pn−m bc
x + y = pm
+ pn−m
= pm
.
b
d
bd
Since n ≥ m, the numerator on the right is an integer. The denominator are relatively prime to p
since b, d are, though the numerator is possibly divisible by p (though only if n = m and p|(ad+bc)).
This means that we can write x + y = pm+k fe where e, f ∈ Z are prime to p and k ≥ 0. Thus
�
�
|x + y|p = p−m−k ≤ p−m = max p−m , p−n = max {|xp |, |yp |}
Notice that our proof shows that we actually have equality |x + y|p = max {|x|p , |y|p } (since
k = 0 above) except possibly in the case |x|p = |y|p .
Exercise 6.12. Find two integers x, y ∈ Z such that
(i) |x|3 = |y|3 = 13 but |x + y|3 = 19 .
(ii) |x|3 = |y|3 = |x + y|3 = 13 .
Proposition 6.2.6. Every ball Br (x) in Qp is both open and closed. Thus the singleton sets in Qp
are closed.
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Using the fact that the balls are closed, one can show that Qp is totally disconnected, i.e., its
connected components are the singleton sets. However the singleton sets are not open, as that would
imply Qp has the discrete topology, i.e., every set would be both open and closed.
Proof. Each ball is open by definition. The following two exercises
� show Br (x) is also closed.
Then for any x ∈ Qp , the intersection of the closed sets r>0 Br (x) = {x}, which must be
closed.
Exercise 6.13. Show Br (x) = x + Br (0) = {x + y : y ∈ Br (0)}.
Exercise 6.14. Show that Br (0) is closed for any r ∈ R.
Your proof of the second exercise should make use of the fact that | · |p is a discrete absolute
value, i.e., the valuation ordp : Qp → R actually has image Z, which is a discrete subset of R. In
other words, the image of | · |p = p−ordp (·) , namely pZ , is discrete in R except for the limit point at
0. On the other hand, the image of the ordinary absolute value | · |∞ on R is a continuous subset of
R, namely R≥0 .
�
Another strange, but nice thing, about analysis on Qp is that a series
xn converges if and
only if xn → 0 in Qp .
While these are some very fundamental differences between R and Qp , you shouldn’t feel that
Qp is too unnatural—just different from what you’re familiar with. To see that Qp isn’t too strange,
observe the following:
Proposition 6.2.7. Qp and R are both Hausdorff and locally compact, but not compact.
Proof. The results for R should be familiar, so we will just show them for Qp .
Recall a space is Hausdorff if any two points can be separated by open sets. Qp is Hausdorff
since it is a metric space: namely if x �= y ∈ Qp , let d = d(x, y) = |x + y|p . Then for r ≤ d2 , Br (x)
and Br (y) are open neighborhoods of x and y which are disjoint.
Recall a Hausdorff space is locally compact if every point has a compact neighborhood. Around
any x ∈ Qp , we can take the closed ball B r (x) of radius r. This is a closed and (totally) bounded
set in a complete metric space, and therefore compact. (In fact one could also take the open ball
B r (x), since we know it is closed from the previous exercise.)
Perhaps more instructively, one can show B r (x) is sequentially compact in Qp , which is equivalent to compactness being a metric space. We may take a specific r if we want, say r = 1. Further
since B 1 (x) = x + B 1 (0) by the exercise above, it suffices to show B 1 (0) = {x ∈ Qp : |x|p ≤ 1} = Zp
is sequentially compact. If
x1 = a10 + a11 p + a12 p2 + · · ·
x2 = a20 + a21 p + a22 p2 + · · ·
x3 = a30 + a31 p + a32 p2 + · · ·
..
.
is a Cauchy sequence, then for any � > 0, there is an N ∈ N such that |xm − xn |p < � for all
m, n > N . Take � = p−r for r > 0. Then |xm − xn |p < � = p−r means xm ≡ xn mod pr+1 , i.e.,
the coefficients of 1, p, p2 , . . . , pr must be the same for all xm , xn with m, n > N . Let a0 , a1 , . . . , ar
93
denote these coefficients. We can do this for larger and larger r (note that a0 , . . . , ar−1 will never
change) to get a sequence (an ), and then it is clear that the above sequence converges to
x = a0 + a1 p + a2 p2 + · · · ∈ Zp .
This provides a second proof of local compactness.
To see Qp is not compact, observe the sequence x1 = p−1 , x2 = p−2 , x3 = p−3 , . . . has no
convergent subsequence. Geometrically, |xn | = pn , so this is a sequence of points getting further
and further from 0, and the distance to 0 goes to infinity.
We remark that Q, with either usual subspace topology coming from R or the one coming from
Qp , is a space which is not locally compact. The reason is any open neighborhood about a point is
not complete—the limit points are contained in the completion of Q (w.r.t. to whichever absolute
value we are considering), but not in Q. (The trivial absolute value |·|0 induces the discrete topology
on Q, meaning single points are open sets, so it is trivially locally compact.)
The general definition of a local field is a locally compact field, hence we see that Qp and R
are local fields, whereas Q (with the usual topology) is not.
94